To analyze whether the set $N$ of positive integers, under the relation $m \leq n$ meaning that $m$ divides $n$, is a lattice and whether it is complete, let's break it down step-by-step.

1. Is $N$ a lattice?

A lattice is a partially ordered set (poset) where every two elements have a supremum (least upper bound) and an infimum (greatest lower bound). In your case, we are dealing with the poset of integers with the divisibility relation.

• Supremum (Least Upper Bound): For any two elements $m$ and $n$ in $N$, the least upper bound would be the least common multiple (LCM) of $m$ and $n$, since $\text{lcm}(m, n)$ is the smallest number that is divisible by both $m$ and $n$.

• Infimum (Greatest Lower Bound): The greatest lower bound would be the greatest common divisor (GCD) of $m$ and $n$, as $\text{gcd}(m, n)$ is the largest number that divides both $m$ and $n$.

Therefore, every pair of positive integers $m$ and $n$ in $N$ has both a least upper bound (LCM) and a greatest lower bound (GCD). This means that the set $N$ with the divisibility relation is a lattice.

2. Is $N$ complete?

A poset is complete if every subset of the poset has both a supremum (join) and an infimum (meet).

For the set $N$, under the divisibility relation:

• Supremum (Join) of a Subset: The least upper bound of any subset of $N$ would be the least common multiple (LCM) of all the elements in the subset. The LCM exists for any finite subset of $N$, but the situation becomes more complicated when considering infinite subsets.

• Infimum (Meet) of a Subset: The greatest lower bound of any subset of $N$ would be the greatest common divisor (GCD) of all the elements in the subset. Similarly, the GCD exists for finite subsets but may not exist for infinite subsets.

The critical issue is whether the GCD and LCM always exist for every subset, including infinite subsets. The GCD of an infinite set of positive integers might not be well-defined if the integers do not share any common divisors (e.g., the set of all prime numbers has no greatest common divisor). Similarly, the LCM of an infinite set could potentially not exist if there is no finite least common multiple that bounds all elements.

Thus, $N$ is not complete because not every subset of $N$ will necessarily have a least upper bound or greatest lower bound (particularly for infinite subsets).

Summary:

• Is $N$ a lattice? Yes, because every pair of integers has both a least upper bound (LCM) and a greatest lower bound (GCD).
• Is $N$ complete? No, because not every subset of $N$ (particularly infinite subsets) has a least upper bound or greatest lower bound.

Would you like further details or explanations on any part of this? Here are some related questions:

1. What is the definition of a lattice in more general terms?
2. How does the least common multiple (LCM) relate to divisibility?
3. Can an infinite subset of integers have a well-defined greatest common divisor?
4. What are some other examples of lattices outside of the divisibility relation?
5. How does the concept of completeness relate to the existence of suprema and infima in a poset?

Tip: When working with divisibility relations, remember that the GCD and LCM are key tools for finding the infimum and supremum of two elements in the poset.